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課程名稱︰微積分甲
課程性質︰系必帶
課程教師︰林紹雄
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰98/1/11
考試時限（分鐘）：180分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
There are problems A to H with a total of 140 points. Please write down your
computational or proof steps clearly on the answer sheets.
A.Find the following anti-derivatives, or definite integrals, or determine the
convergence of the improper integrals. Each has 9 points.
(a)∫(0 to 1) [(x^4)(1-x)^4/(x^2)+1]dx
(b)∫(1+e^x)^(1/2)
(c)∫dx/x(1+x+x^2)^(1/2)
(d)∫(0 to π/2) secx dx/2tanx+secx-1
B.(20 points) The curve y^2+x(x+1/3)^2=0 enclosed a bounded plane region. Find
its area, its centroid, and the arclength of its boundary curve. Now we
revolve this plane region around the y-axis to obtain a solid of revolution.
Find the volume and the surface area of this solid.
C.(10 points) Apply the Taylor expansion with remainder to estimate
∫(0 to 0.25) (1+x^2)^(1/3)dx accurately up to three decimal points.
D.(12 points) A cat chase a mouse along a straight line. Their distance at
time t is d(t). The cat keeps a constant velocity 1, while the mouse's
velocity at time t is 1/d(t). Write down the differential equation satisfied
by d(t), and find d(t). Show that if d(0)>1, then d(t)>1 for all t>0 so that
the cat can never be closer to the mouse than distance 1.
E. Determine which of the following infinite series converges absolutely, or
converges conditionallly, or diverges. Each has 6 points.
(a)Σ(0,∞) (-1)^n (n^100 *2^n / (n!)^(1/2))
(b)a - b/2 + a/3 - b/4 + a/5 - b/6 + ....
a and b in (b) are two positive constants.
F.(12 points) Find the radius of convergence R for the power series
Σ(1,∞)[(b^n / n^2)+(a^n / n )] x^n ,
and discuss the convergent behaviour of the series at x=±R. Find the sum of
this series.
G.(13 points) Let μ(n) and υ(n) (n=1,2,3,...) be two sequences.
s(n)=Σ(1,n)υ(j) is the partial sum of Σ(1,∞)υ(n). If
(a)υ(n) is positive, decreasing and convergent to 0, and
(b)|s(n)|≦K for all n, where K＞0 is a constant,
prove that Σ(1,∞)μ(n)υ(n) converges.
H.Determine which of the following statements is true. Prove your answer.
Each has 5 points.
(a)If ∫(a to b) |f(x)|dx exists for a given function defined on [a,b], then
∫(a to b) f(x)dx exists also.
(b)Let Sn be the Simpson rule approximation to the integral ∫(0 to 1)f(x)dx
using 2n equal partition on [0,1]. Define the error En=|∫(0 to 1)f(x)dx - Sn|.
If f(x)=x(x-1)(x+1), then En=0
(c)Let f(x) be strictly increasing for all x 屬於 R. If f(0)=0, then there
must exist a real number x(0)≠0 such that the sequence x(n) (n=0,1,2,...)
obtained by applying the Newton method to f(x) starting from x(0) converges
to 0.
(d)If the series Σ(1,∞)a(n) and Σ(1,∞)(-1)^n a(n) converges, then
Σ(1,∞)a(n) converges absolutely.
(e) Since lim(θ→π) 1/θ-π = ±∞, the curve r =1/θ-π on the plane has
the line θ=π as its asymptotic line.
若有符號不清楚的地方，這邊有圖 http://www.badongo.com/pic/5049643
至於答案....不要問我Q_Q
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